Questions — Edexcel S2 (494 questions)

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Edexcel S2 2007 January Q5
5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).
  1. Write down the probability density function of \(X\), for all \(x\).
  2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
  3. \(\mathrm { E } ( X )\),
  4. the standard deviation of \(X\),
  5. the probability that the shorter piece of wire is at most 30 cm long.
Edexcel S2 2007 January Q6
6. Past records from a large supermarket show that \(20 \%\) of people who buy chocolate bars buy the family size bar. On one particular day a random sample of 30 people was taken from those that had bought chocolate bars and 2 of them were found to have bought a family size bar.
  1. Test at the \(5 \%\) significance level, whether or not the proportion \(p\), of people who bought a family size bar of chocolate that day had decreased. State your hypotheses clearly. The manager of the supermarket thinks that the probability of a person buying a gigantic chocolate bar is only 0.02 . To test whether this hypothesis is true the manager decides to take a random sample of 200 people who bought chocolate bars.
  2. Find the critical region that would enable the manager to test whether or not there is evidence that the probability is different from 0.02 . The probability of each tail should be as close to \(2.5 \%\) as possible.
  3. Write down the significance level of this test.
Edexcel S2 2007 January Q7
7. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
2 x ^ { 2 } - x ^ { 3 } , & 0 \leqslant x \leqslant 1
1 , & x > 1 \end{cases}$$
  1. Find \(\mathrm { P } ( X > 0.3 )\).
  2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
  3. Find the probability density function \(\mathrm { f } ( x )\).
  4. Evaluate \(\mathrm { E } ( X )\).
  5. Find the mode of \(X\).
  6. Comment on the skewness of \(X\). Justify your answer.
Edexcel S2 2008 January Q1
  1. (a) Explain what you understand by a census.
Each cooker produced at GT Engineering is stamped with a unique serial number. GT Engineering produces cookers in batches of 2000. Before selling them, they test a random sample of 5 to see what electric current overload they will take before breaking down.
(b) Give one reason, other than to save time and cost, why a sample is taken rather than a census.
(c) Suggest a suitable sampling frame from which to obtain this sample.
(d) Identify the sampling units.
Edexcel S2 2008 January Q2
2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are
  1. exactly 2 faulty bolts,
  2. more than 3 faulty bolts. These bolts are sold in bags of 20. John buys 10 bags.
  3. Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.
Edexcel S2 2008 January Q3
3. (a) State two conditions under which a Poisson distribution is a suitable model to use in statistical work. The number of cars passing an observation point in a 10 minute interval is modelled by a Poisson distribution with mean 1.
(b) Find the probability that in a randomly chosen 60 minute period there will be
  1. exactly 4 cars passing the observation point,
  2. at least 5 cars passing the observation point. The number of other vehicles, other than cars, passing the observation point in a 60 minute interval is modelled by a Poisson distribution with mean 12.
    (c) Find the probability that exactly 1 vehicle, of any type, passes the observation point in a 10 minute period.
Edexcel S2 2008 January Q4
  1. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c l } 0 & y < 1
k \left( y ^ { 4 } + y ^ { 2 } - 2 \right) & 1 \leqslant y \leqslant 2
1 & y > 2 \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 18 }\).
  2. Find \(\mathrm { P } ( Y > 1.5 )\).
  3. Specify fully the probability density function f(y).
Edexcel S2 2008 January Q5
  1. Dhriti grows tomatoes. Over a period of time, she has found that there is a probability 0.3 of a ripe tomato having a diameter greater than 4 cm . She decides to try a new fertiliser. In a random sample of 40 ripe tomatoes, 18 have a diameter greater than 4 cm . Dhriti claims that the new fertiliser has increased the probability of a ripe tomato being greater than 4 cm in diameter.
Test Dhriti's claim at the 5\% level of significance. State your hypotheses clearly.
Edexcel S2 2008 January Q6
6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.
  1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
    1. a Poisson approximation,
    2. a Normal approximation.
  2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.
Edexcel S2 2008 January Q7
  1. (a) Explain what you understand by
    1. a hypothesis test,
    2. a critical region.
    During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
    (b) Find the critical region for a two-tailed test of the hypothesis that the number of incoming calls occurs at a rate of 0.45 per 1 minute interval. The probability in each tail should be as close to \(2.5 \%\) as possible.
    (c) Write down the actual significance level of the above test. In the school holidays, 1 call occurs in a 10 minute interval.
    (d) Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of incoming calls is less during the school holidays than in term time.
Edexcel S2 2008 January Q8
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Write down the mode of \(X\). Find
  3. \(\mathrm { E } ( X )\),
  4. the median of \(X\).
  5. Comment on the skewness of this distribution. Give a reason for your answer.
Edexcel S2 2009 January Q1
  1. A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field.
Find the probability that, in a randomly chosen square there will be
  1. more than 2 daisies,
  2. either 5 or 6 daisies. The botanist decides to count the number of daisies, \(x\), in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x ^ { 2 } = 1386$$
  3. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places.
  4. Explain how the answers from part (c) support the choice of a Poisson distribution as a model.
  5. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square.
Edexcel S2 2009 January Q2
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 2,7 ]\).
    1. Write down fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
    2. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\).
    Find
  2. \(\mathrm { E } \left( X ^ { 2 } \right)\),
  3. \(\mathrm { P } ( - 0.2 < X < 0.6 )\).
Edexcel S2 2009 January Q3
3. A single observation \(x\) is to be taken from a Binomial distribution \(\mathrm { B } ( 20 , p )\). This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
  1. Using a \(5 \%\) level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to \(2.5 \%\).
  2. State the actual significance level of this test. The actual value of \(x\) obtained is 3 .
  3. State a conclusion that can be drawn based on this value giving a reason for your answer.
Edexcel S2 2009 January Q4
4. The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c l } k t & 0 \leqslant t \leqslant 10
0 & \text { otherwise } \end{array} \right.$$
  1. Show that the value of \(k\) is \(\frac { 1 } { 50 }\).
  2. Find \(\mathrm { P } ( T > 6 )\).
  3. Calculate an exact value for \(\mathrm { E } ( T )\) and for \(\operatorname { Var } ( T )\).
  4. Write down the mode of the distribution of \(T\). It is suggested that the probability density function, \(\mathrm { f } ( t )\), is not a good model for \(T\).
  5. Sketch the graph of a more suitable probability density function for \(T\).
Edexcel S2 2009 January Q5
  1. A factory produces components of which \(1 \%\) are defective. The components are packed in boxes of 10 . A box is selected at random.
    1. Find the probability that the box contains exactly one defective component.
    2. Find the probability that there are at least 2 defective components in the box.
    3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
    4. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
      1. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
      2. State the minimum number of visits required to obtain a significant result.
    6. State an assumption that has been made about the visits to the server.
    In a random two minute period on a Saturday the web server is visited 20 times.
  2. Using a suitable approximation, test at the \(10 \%\) level of significance, whether or not the rate of visits is greater on a Saturday.
Edexcel S2 2009 January Q7
  1. A random variable \(X\) has probability density function given by
$$f ( x ) = \left\{ \begin{array} { c l } - \frac { 2 } { 9 } x + \frac { 8 } { 9 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$
  1. Show that the cumulative distribution function \(\mathrm { F } ( x )\) can be written in the form \(a x ^ { 2 } + b x + c\), for \(1 \leqslant x \leqslant 4\) where \(a , b\) and \(c\) are constants.
  2. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. Given that the median of \(X\) is 1.88
  4. describe the skewness of the distribution. Give a reason for your answer.
Edexcel S2 2010 January Q1
  1. A manufacturer supplies DVD players to retailers in batches of 20 . It has \(5 \%\) of the players returned because they are faulty.
    1. Write down a suitable model for the distribution of the number of faulty DVD players in a batch.
    Find the probability that a batch contains
  2. no faulty DVD players,
  3. more than 4 faulty DVD players.
  4. Find the mean and variance of the number of faulty DVD players in a batch.
Edexcel S2 2010 January Q2
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2
\frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4
1 , & x > 4 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 0 )\).
  2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Write down the name of the distribution of \(X\).
  4. Find the mean and the variance of \(X\).
  5. Write down the value of \(\mathrm { P } ( X = 1 )\).
Edexcel S2 2010 January Q3
  1. A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.
    1. Find the probability that it will work continuously for 5 hours without a breakdown.
    Find the probability that, in an 8 hour period,
  2. the robot will break down at least once,
  3. there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
  4. Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.
Edexcel S2 2010 January Q4
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3
3 k & 3 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 9 }\)
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the mean of \(X\).
  4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)
Edexcel S2 2010 January Q5
  1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
Find the probability that
  1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
  2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
Edexcel S2 2010 January Q6
6. (a) Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
(b) Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
(c) Write down the actual significance level of the test. The value of the observation was found to be 15 .
(d) Comment on this finding in light of your critical region.
Edexcel S2 2010 January Q7
  1. A bag contains a large number of coins. It contains only \(1 p\) and \(2 p\) coins in the ratio \(1 : 3\)
    1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of the values of this population of coins.
    A random sample of size 3 is taken from the bag.
  2. List all the possible samples.
  3. Find the sampling distribution of the mean value of the samples.
Edexcel S2 2011 January Q1
  1. A disease occurs in \(3 \%\) of a population.
    1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution.
    2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.
    3. Find the mean and variance of the number of people with the disease in a random sample of 100 people.
    A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.
  2. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.